<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2900.2963" name=GENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<DIV><FONT face="Courier New" size=2>(Best viewed in fixed width)</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New" size=2>Let f be defined as</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New" size=2>f(x, y) = <BR> 0, if y
> x<BR> 1, if y = 0<BR> 2*SUM(k >= 1
and x-k^2 >= y; f(x-k^2, y-1)), otherwise.</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New" size=2>A table of f(x, y), omitting the zero
elements:</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New" size=2> \y|
0 1 2 3
4 5 6 7
8 9 10<BR>
x\|<BR>-----+-------------------------------------------------------<BR>
0 | 1<BR> 1 |
1 2<BR> 2 | 1
2 4<BR> 3 | 1
2 4 8<BR> 4 |
1 4 4 8
16<BR> 5 | 1 4
12 8 16 32<BR> 6
| 1 4 12
32 16 32 64<BR> 7
| 1 4 12
32 80 32 64 128<BR> 8
| 1 4 16
32 80 192 64 128 256<BR> 9
| 1 6 16
56 80 192 448 128 256 512<BR> 10
| 1 6 24 56
176 192 448 1024 256 512 1024</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New" size=2>Now use the xth row of this table as
differences to generate sequence S_x. </FONT><FONT face="Courier New"
size=2>For example, taking x = 3, the third row is (1 2 4 8). Using these as
differences, we generate the sequence:</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New"
size=2>
8 8 8
...<BR>
4 12 20
28
...<BR>
2 6 18
38 66 ...<BR> S_3 =
1 3 9
27 65 131 ...</FONT></DIV>
<DIV><FONT face="Courier New" size=2></FONT> </DIV>
<DIV><FONT face="Courier New" size=2>S_3 is indexed starting at 0.</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New" size=2>It appears that S_x(n) gives the number of
points in Z^n with norm <= sqrt(x). For example, there are S_3(4) =
65 points of Z^4 norm <= sqrt(3), namely:</FONT></DIV>
<DIV><FONT face="Courier New" size=2></FONT> </DIV>
<DIV><FONT face="Courier New" size=2><FONT face=Arial>(-1 -1 -1 0)<BR>(-1 -1 0
-1)<BR>(-1 -1 0 0)<BR>(-1 -1 0 1)<BR>(-1 -1 1 0)<BR>(-1 0 -1 -1)<BR>(-1 0 -1
0)<BR>(-1 0 -1 1)<BR>(-1 0 0 -1)<BR>(-1 0 0 0)<BR>(-1 0 0 1)<BR>(-1 0 1
-1)<BR>(-1 0 1 0)<BR>(-1 0 1 1)<BR>(-1 1 -1 0)<BR>(-1 1 0 -1)<BR>(-1 1 0
0)<BR>(-1 1 0 1)<BR>(-1 1 1 0)<BR>(0 -1 -1 -1)<BR>(0 -1 -1 0)<BR>(0 -1 -1
1)<BR>(0 -1 0 -1)<BR>(0 -1 0 0)<BR>(0 -1 0 1)<BR>(0 -1 1 -1)<BR>(0 -1 1 0)<BR>(0
-1 1 1)<BR>(0 0 -1 -1)<BR>(0 0 -1 0)<BR>(0 0 -1 1)<BR>(0 0 0 -1)<BR>(0 0 0
0)<BR>(0 0 0 1)<BR>(0 0 1 -1)<BR>(0 0 1 0)<BR>(0 0 1 1)<BR>(0 1 -1 -1)<BR>(0 1
-1 0)<BR>(0 1 -1 1)<BR>(0 1 0 -1)<BR>(0 1 0 0)<BR>(0 1 0 1)<BR>(0 1 1 -1)<BR>(0
1 1 0)<BR>(0 1 1 1)<BR>(1 -1 -1 0)<BR>(1 -1 0 -1)<BR>(1 -1 0 0)<BR>(1 -1 0
1)<BR>(1 -1 1 0)<BR>(1 0 -1 -1)<BR>(1 0 -1 0)<BR>(1 0 -1 1)<BR>(1 0 0 -1)<BR>(1
0 0 0)<BR>(1 0 0 1)<BR>(1 0 1 -1)<BR>(1 0 1 0)<BR>(1 0 1 1)<BR>(1 1 -1 0)<BR>(1
1 0 -1)<BR>(1 1 0 0)<BR>(1 1 0 1)<BR>(1 1 1 0)</FONT></DIV>
<DIV><BR></DIV></FONT></BODY></HTML>